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How to find generalized eigenvectors

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I have difficulties understanding an easy method to find the generalized eigenvectors of a nilpotent matrix.
For example, for the matrix A= 1st : 1 0 0
2nd; -1 2 0
3rd: 1 1 2

The first eigenvalue is 1, and the other is 2 with multiplicity 2.
I know how to find the first eigenvector for eigenvalue 1 which is v1= (1,1, -2)^T, but I got difficulty to find the two other ones. The book gives for v2= (0,0, 1)^T, and for V3, he added to use formula (A-2I)^2 v =0 to find v3 to be v3= (0,1,0)^T.

I tried to follow but I cannot find how to get v3.

( I have the complete solution for the IVP, all I need are the eigenvectors).

Please explain this to me!

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If an eigenvalue has multiplicity m there might not be a set of linearly independent ...

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